it explains matrix multiplication. its algorithm and analysis

2. Strassen's Matrix Multiplication
Presented by:
Ali Mamoon
07-0014

3. Contents
 Matrix multiplication
 Divide and Conquer
 Strassen's idea
 Analysis

5. Standard algorithm
for i ←1 to n
do for j←1 to n
do cij←0
for k←1 to n
do cij←cij+ aik⋅bkj
N
C i, j a i ,k bk , j
k 1
N N N
3 3
Thus T ( N ) c cN O(N )
i1 j1k 1

6. Divide-and-Conquer
 Divide-and conquer is a general algorithm design
paradigm:
 Divide: divide the input data S in two or more disjoint
subsets S1, S2, …
 Recur: solve the sub problems recursively
 Conquer: combine the solutions for S1, S2, …, into a
solution for S
 The base case for the recursion are sub problems of
constant size
 Analysis can be done using recurrence equations

7. Matrix Multiplication through
Divide and Conquer Approach

8. Divide-and-conquer algorithm

10. Master theorem

12. Strassen’s idea
 Multiply 2×2matrices with only 7 recursive multiplications.
 P1= a⋅(f–h) r=P5+ P4–P2+ P6
P2= (a+ b) ⋅h
 s=P1+ P2
P3= (c+ d) ⋅e
 t=P3+ P4
 P4= d⋅(g–e) u=P5+ P1–P3–P7
P5= (a+ d) ⋅(e+ h)

P6= (b–d) ⋅(g+ h)

P7= (a–c) ⋅(e+ f )

 Note:
 7mults, 18adds/subs.
 Note: No reliance on commutative of multiplication!

14. Strassen Algorithm
void matmul(int *A, int *B, int *R, int n) {
if (n == 1) {
(*R) += (*A) * (*B);
} else {
matmul(A, B, R, n/4);
matmul(A, B+(n/4), R+(n/4), n/4);
matmul(A+2*(n/4), B, R+2*(n/4), n/4);
matmul(A+2*(n/4), B+(n/4), R+3*(n/4), n/4);
matmul(A+(n/4), B+2*(n/4), R, n/4);
matmul(A+(n/4), B+3*(n/4), R+(n/4), n/4);
matmul(A+3*(n/4), B+2*(n/4), R+2*(n/4), n/4);
matmul(A+3*(n/4), B+3*(n/4), R+3*(n/4), n/4); Divide matrices in
} sub-matrices and
recursively multiply
sub-matrices

15. Analysis of Strassen's
 T(n) = 7T(n/2) + Θ(n2)
 nlogba= nlog27≈n2.81
 ⇒CASE1⇒T(n) = Θ(nlg7).
 The number2.81may not seem much smaller than 3, but
because the difference is in the exponent, the impact on
running time is significant. In fact, Strassen’s algorithm
beats the ordinary algorithm on today’s machines for
n≥32or so.
 Best to date (of theoretical interest only): Θ(n2.376…)